The basic physics of circular motion require 2 directions of motion, with vectors of 90 deg. to each other. In orbit, the forward vector is provided by the initial propellant burn that got you into orbit. When you are in a circular orbit, this force is no longer providing acceleration, so you have a steady velocity (see Newton's first law). Gravity is the vector at 90 degrees. It is providing constant acceleration in the direction of the center of mass of the planet. Of course, since this is a stationary point from the perspective of the orbiter, the direction of the pull keeps changing, since they are also moving along the other vector. These two directions of motion are balanced such that the resulting path curves into a circle. Orbits can essentially be considered constantly falling, but moving forward just fast enough that you keep missing the point you are falling towards by a constant amount.
With a pro-grade burn, you are adding acceleration to the forward plane of this two vector system, essentially increasing the forward vector in relation to the downward vector. Since the balance of the vectors has changed, the resulting path will move closer to a straight line the more forward velocity you add. While you will initially move closer to your target because you appear to be moving in the same direction, you are now actually accelerating on both vectors, while your target continues to accelerate on only one. Your paths will quickly diverge, as your increased forward motion moves you in a flatter trajectory, while the other object maintains its curved, circular path.
To get closer to an object while remaining in the same orbit, you would have to increase the gravity acting on you at the same time you are increasing your forward velocity. This could be simulated by adding a thruster to the top of your orbiting vehicle, and applying a downward force as you accelerated forward. However, the vertical thrust would have to remain constant, even after you cut forward acceleration, to simulate the increase in constant gravitic acceleration you would need to remain in that orbit at the higher forward velocity. Most current orbiter vehicles could not carry the amount of fuel necessary for the type of constant burn this would require. Hence the far more fuel efficient use of altered orbits described in the answers above to catch up to another object.